What would happen if a poked someone with a million-mile-long stick?

Suppose I was holding the end of a million-mile-long stick and the other end was an inch from another person's body. If nothing can travel faster than light, then what would happen if I thrust the stick forward? Would it hit the person right away or would it take a few seconds to?

The stick is not a perfectly rigid object. There is no such thing as a perfectly rigid object - relativity forbids it.

The stick is made of atoms. Any acceleration at one point in a stick propagates through the stick at the speed of sound of whatver material it is made of. If the stick were made of diamond, its speed of sound would be about 8 miles/s. So your push would take almost a day and a half to reach the other end.